On the Hofer–Zehnder Capacity of Twisted Tangent Bundles

Abstract

In this thesis, we deal with the Hofer–Zehnder capacity of disc subbundles of twisted tangent bundles. While in the literature for most cases only the finiteness of this capacity is shown, we use symmetries to determine exact values of the capacity. We therefore restrict ourselves to a class of homogeneous Kähler manifolds, called Hermitian symmetric spaces. For these, we construct a symplectomorphism that identifies the twisted tangent bundle, or at least a neighborhood of the zero section that we can specify explicitly, and the Hermitian tangent bundle. The advantage of the Hermitian tangent bundle is that the fibers are symplectic. This makes it easier to study holomorphic curves, which we use to obtain an upper bound on the Hofer–Zehnder capacity. We get the lower bound by specifying a Hamiltonian that generates a circle action. The oscillation of such a Hamiltonian always yields a lower bound.
We also clarify the relationship between the twisted, respectively Hermitian, symplectic structure to the hyperkähler structure in a neighborhood of the zero section of the tangent bundle of a Hermitian symmetric space.

For various reasons, it is much harder to determine the Hofer–Zehnder capacity for standard tangential bundles than it is for the twisted case. Nevertheless, we were able to compute the Hofer–Zehnder capacity for the disc subbundle of the standard tangent bundle of the complex projective space $\mathbb{C}P^n$ and the real projective space $\mathbb{R}P^n$. To obtain the lower bound it is for the former sufficient to consider the kinetic Hamiltonian, i.e. geodesic flow, while in the second case, geodesic billiards must be used. For the upper bound one uses the symmetries of the spaces to show that the disc subbundle of the tangent bundles compactify to the product of two complex projective spaces $\mathbb{C}P^n × \mathbb{C}P^n$ in the first case and the complex projective space $\mathbb{C}P^n$ in the second case. In these compact symplectic manifolds one can again study holomorphic spheres in order to construct upper bounds. In fact, we also show in the twisted case that the disc subbundle of the tangent bundle of the complex projective space compactifies to the product, but now with differently weighted factors.

Furthermore, this thesis includes the computation of the Hofer–Zehnder capacity of  Hermitian symmetric spaces of compact type. This exploits the fact that Hermitian symmetric spaces can be represented as coadjoint orbits. In this representation it is relatively easy to specify a Hamiltonian which generates a semi-free circle action and which attains its minimum at an isolated point. The oscillation of such a Hamiltonian provides both lower and upper bounds and thus determines the capacity.